Theorem mulsrpr 9344

Description: Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)

Assertion

Ref	Expression
mulsrpr	|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R )

Proof of Theorem mulsrpr

Dummy variables x y z w v u t s f g h a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 4654	. 2 |- <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. e. _V
2		opex 4654	. 2 |- <. ( ( a .P. g ) +P. ( b .P. h ) ) , ( ( a .P. h ) +P. ( b .P. g ) ) >. e. _V
3		opex 4654	. 2 |- <. ( ( c .P. t ) +P. ( d .P. s ) ) , ( ( c .P. s ) +P. ( d .P. t ) ) >. e. _V
4		enrex 9338	. 2 |- ~R e. _V
5		enrer 9336	. 2 |- ~R Er ( P. X. P. )
6		df-enr 9327	. 2 |- ~R = { <. x , y >. | ( ( x e. ( P. X. P. ) /\ y e. ( P. X. P. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z +P. u ) = ( w +P. v ) ) ) }
7		oveq12 6199	. . . 4 |- ( ( z = a /\ u = d ) -> ( z +P. u ) = ( a +P. d ) )
8		oveq12 6199	. . . 4 |- ( ( w = b /\ v = c ) -> ( w +P. v ) = ( b +P. c ) )
9	7, 8	eqeqan12d 2474	. . 3 |- ( ( ( z = a /\ u = d ) /\ ( w = b /\ v = c ) ) -> ( ( z +P. u ) = ( w +P. v ) <-> ( a +P. d ) = ( b +P. c ) ) )
10	9	an42s 823	. 2 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> ( ( z +P. u ) = ( w +P. v ) <-> ( a +P. d ) = ( b +P. c ) ) )
11		oveq12 6199	. . . 4 |- ( ( z = g /\ u = s ) -> ( z +P. u ) = ( g +P. s ) )
12		oveq12 6199	. . . 4 |- ( ( w = h /\ v = t ) -> ( w +P. v ) = ( h +P. t ) )
13	11, 12	eqeqan12d 2474	. . 3 |- ( ( ( z = g /\ u = s ) /\ ( w = h /\ v = t ) ) -> ( ( z +P. u ) = ( w +P. v ) <-> ( g +P. s ) = ( h +P. t ) ) )
14	13	an42s 823	. 2 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( ( z +P. u ) = ( w +P. v ) <-> ( g +P. s ) = ( h +P. t ) ) )
15		df-mpr 9326	. 2 |- .pR = { <. <. x , y >. , z >. | ( ( x e. ( P. X. P. ) /\ y e. ( P. X. P. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .P. u ) +P. ( v .P. f ) ) , ( ( w .P. f ) +P. ( v .P. u ) ) >. ) ) }
16		oveq12 6199	. . . . 5 |- ( ( w = a /\ u = g ) -> ( w .P. u ) = ( a .P. g ) )
17		oveq12 6199	. . . . 5 |- ( ( v = b /\ f = h ) -> ( v .P. f ) = ( b .P. h ) )
18	16, 17	oveqan12d 6209	. . . 4 |- ( ( ( w = a /\ u = g ) /\ ( v = b /\ f = h ) ) -> ( ( w .P. u ) +P. ( v .P. f ) ) = ( ( a .P. g ) +P. ( b .P. h ) ) )
19	18	an4s 822	. . 3 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> ( ( w .P. u ) +P. ( v .P. f ) ) = ( ( a .P. g ) +P. ( b .P. h ) ) )
20		oveq12 6199	. . . . 5 |- ( ( w = a /\ f = h ) -> ( w .P. f ) = ( a .P. h ) )
21		oveq12 6199	. . . . 5 |- ( ( v = b /\ u = g ) -> ( v .P. u ) = ( b .P. g ) )
22	20, 21	oveqan12d 6209	. . . 4 |- ( ( ( w = a /\ f = h ) /\ ( v = b /\ u = g ) ) -> ( ( w .P. f ) +P. ( v .P. u ) ) = ( ( a .P. h ) +P. ( b .P. g ) ) )
23	22	an42s 823	. . 3 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> ( ( w .P. f ) +P. ( v .P. u ) ) = ( ( a .P. h ) +P. ( b .P. g ) ) )
24	19, 23	opeq12d 4165	. 2 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> <. ( ( w .P. u ) +P. ( v .P. f ) ) , ( ( w .P. f ) +P. ( v .P. u ) ) >. = <. ( ( a .P. g ) +P. ( b .P. h ) ) , ( ( a .P. h ) +P. ( b .P. g ) ) >. )
25		oveq12 6199	. . . . 5 |- ( ( w = c /\ u = t ) -> ( w .P. u ) = ( c .P. t ) )
26		oveq12 6199	. . . . 5 |- ( ( v = d /\ f = s ) -> ( v .P. f ) = ( d .P. s ) )
27	25, 26	oveqan12d 6209	. . . 4 |- ( ( ( w = c /\ u = t ) /\ ( v = d /\ f = s ) ) -> ( ( w .P. u ) +P. ( v .P. f ) ) = ( ( c .P. t ) +P. ( d .P. s ) ) )
28	27	an4s 822	. . 3 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> ( ( w .P. u ) +P. ( v .P. f ) ) = ( ( c .P. t ) +P. ( d .P. s ) ) )
29		oveq12 6199	. . . . 5 |- ( ( w = c /\ f = s ) -> ( w .P. f ) = ( c .P. s ) )
30		oveq12 6199	. . . . 5 |- ( ( v = d /\ u = t ) -> ( v .P. u ) = ( d .P. t ) )
31	29, 30	oveqan12d 6209	. . . 4 |- ( ( ( w = c /\ f = s ) /\ ( v = d /\ u = t ) ) -> ( ( w .P. f ) +P. ( v .P. u ) ) = ( ( c .P. s ) +P. ( d .P. t ) ) )
32	31	an42s 823	. . 3 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> ( ( w .P. f ) +P. ( v .P. u ) ) = ( ( c .P. s ) +P. ( d .P. t ) ) )
33	28, 32	opeq12d 4165	. 2 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> <. ( ( w .P. u ) +P. ( v .P. f ) ) , ( ( w .P. f ) +P. ( v .P. u ) ) >. = <. ( ( c .P. t ) +P. ( d .P. s ) ) , ( ( c .P. s ) +P. ( d .P. t ) ) >. )
34		oveq12 6199	. . . . 5 |- ( ( w = A /\ u = C ) -> ( w .P. u ) = ( A .P. C ) )
35		oveq12 6199	. . . . 5 |- ( ( v = B /\ f = D ) -> ( v .P. f ) = ( B .P. D ) )
36	34, 35	oveqan12d 6209	. . . 4 |- ( ( ( w = A /\ u = C ) /\ ( v = B /\ f = D ) ) -> ( ( w .P. u ) +P. ( v .P. f ) ) = ( ( A .P. C ) +P. ( B .P. D ) ) )
37	36	an4s 822	. . 3 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( ( w .P. u ) +P. ( v .P. f ) ) = ( ( A .P. C ) +P. ( B .P. D ) ) )
38		oveq12 6199	. . . . 5 |- ( ( w = A /\ f = D ) -> ( w .P. f ) = ( A .P. D ) )
39		oveq12 6199	. . . . 5 |- ( ( v = B /\ u = C ) -> ( v .P. u ) = ( B .P. C ) )
40	38, 39	oveqan12d 6209	. . . 4 |- ( ( ( w = A /\ f = D ) /\ ( v = B /\ u = C ) ) -> ( ( w .P. f ) +P. ( v .P. u ) ) = ( ( A .P. D ) +P. ( B .P. C ) ) )
41	40	an42s 823	. . 3 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( ( w .P. f ) +P. ( v .P. u ) ) = ( ( A .P. D ) +P. ( B .P. C ) ) )
42	37, 41	opeq12d 4165	. 2 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> <. ( ( w .P. u ) +P. ( v .P. f ) ) , ( ( w .P. f ) +P. ( v .P. u ) ) >. = <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. )
43		df-mr 9330	. 2 |- .R = { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\ E. a E. b E. c E. d ( ( x = [ <. a , b >. ] ~R /\ y = [ <. c , d >. ] ~R ) /\ z = [ ( <. a , b >. .pR <. c , d >. ) ] ~R ) ) }
44		df-nr 9328	. 2 |- R. = ( ( P. X. P. ) /. ~R )
45		mulcmpblnr 9342	. 2 |- ( ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) ) /\ ( ( g e. P. /\ h e. P. ) /\ ( t e. P. /\ s e. P. ) ) ) -> ( ( ( a +P. d ) = ( b +P. c ) /\ ( g +P. s ) = ( h +P. t ) ) -> <. ( ( a .P. g ) +P. ( b .P. h ) ) , ( ( a .P. h ) +P. ( b .P. g ) ) >. ~R <. ( ( c .P. t ) +P. ( d .P. s ) ) , ( ( c .P. s ) +P. ( d .P. t ) ) >. ) )
46	1, 2, 3, 4, 5, 6, 10, 14, 15, 24, 33, 42, 43, 44, 45	ovec 7310	1 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R .R [ <. C , D >. ] ~R ) = [ <. ( ( A .P. C ) +P. ( B .P. D ) ) , ( ( A .P. D ) +P. ( B .P. C ) ) >. ] ~R )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   <.cop 3981  (class class class)co 6190   [cec 7199   P.cnp 9127   +P. cpp 9129   .P. cmp 9130   .pR cmpr 9133   ~R cer 9134   R.cnr 9135   .R cmr 9140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-omul 7025  df-er 7201  df-ec 7203  df-qs 7207  df-ni 9142  df-pli 9143  df-mi 9144  df-lti 9145  df-plpq 9178  df-mpq 9179  df-ltpq 9180  df-enq 9181  df-nq 9182  df-erq 9183  df-plq 9184  df-mq 9185  df-1nq 9186  df-rq 9187  df-ltnq 9188  df-np 9251  df-plp 9253  df-mp 9254  df-ltp 9255  df-mpr 9326  df-enr 9327  df-nr 9328  df-mr 9330

This theorem is referenced by:  mulclsr  9352  mulcomsr  9357  mulasssr  9358  distrsr  9359  m1m1sr  9361  1idsr  9366  00sr  9367  recexsrlem  9371  mulgt0sr  9373